Cooperative spontaneous emission as a many-body eigenvalue problem

We study emission of a single photon from a spherically symmetric cloud of N atoms (one atom is excited; N-1 are in the ground state) and present an exact analytical expression for eigenvalues and eigenstates of this many-body problem. We found that some states decay much faster than the single-atom decay rate, while other states are trapped and undergo very slow decay. When the size of the atomic cloud is small compared with the radiation wavelength, we found that the radiation frequency undergoes a large shift.